The necklace poset is a symmetric chain order

Let Nn denote the quotient poset of the Boolean lattice, Bn, under the relation equivalence under rotation. Griggs, Killian, and Savage proved that Np is a symmetric chain order for prime p. In this paper, we settle the question posed in that paper, namely whether Nn is a symmetric chain order for all n. This paper provides an algorithm that produces a symmetric chain decomposition (or SCD). We accomplish this by modifying bracketing from Greene and Kleitman. This allows us to take appropriate “middles” of certain chains from the Greene-Kleitman SCD for Bn. We also prove additional properties of the resulting SCD and show that this settles a related conjecture.